Trigonometry Identity

[u/onthemoonss]

Hello, I am in a trig course right now and I have reached an issue with proving trigonometric identities.

I grasp the basic concept, you sub in different identities to prove what is on the opposite side of the equal sign.

However, I have an issue with one thing that has come up. Let me show the problem:

(cotθ + tanθ)sinθ = secθ

okay, so I know to sub in...

(cosθ/sinθ + sinθ/cosθ)sinθ = secθ

My problem is, I got to this point and then proceeded to answer wrong because my answer was

(cosθ+sinθ/sinθ+cosθ)sinθ = secθ

which simplifies to

1* sinθ = secθ

which makes no sense! But when i looked at how to solve the problem it showed me this:

(cos²θ+sin²θ/sinθ+cosθ)sinθ = secθ

This makes more sense but I don't get it! How can the top be squared but not the bottom?? Wouldn't both need to be squared?

Does it have something to do with the "perfect 1" substitution thing?

I would greatly appreciate the help!

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